Metric Measure Spaces with Variable Ricci Bounds and Couplings of Brownian Motions

2014/05/02 by Sturm, Karl-Theodor · 1 citation
#31E #51F #58C #58J #60D #60H #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.1405.0459

Abstract

The goal of this paper is twofold: we study metric measure spaces (X,d,m) with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function k:X→ \mathbb R we introduce the curvature-dimension condition CD(k,∞) which canonically extends the curvature-dimension condition CD(K,∞) of Lott-Sturm-Villani for constant K∈ \mathbb R. For infinitesimally Hilbertian spaces we prove i) its equivalence to an evolution-variation inequality EVI_k which in turn extends the EVI_K-inequality of Ambrosio-Gigli-Savaré; ii) its stability under convergence and its local-to-global property. For metric measure spaces with uniform lower curvature bounds K we prove that for each pair of initial distributions μ₁,μ₂ on X there exists a coupling B_t=(B_t¹,B_t²), t≥0, of two Brownian motions on X with the given initial distributions such that a.s. for all s,t≥0 d(B¹_s+t,B²_s+t)≤ e^{-K t/2}⋅ d(B_s¹,B_s²).

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